Does Lady Russell know he is in Bath? There is no proof, but I would say it was much more likely that she did know than that she didn't, if he had been in the city for two days.

We are told that 'everybody was so liable to meet everybody in Bath', and in fact, Anne did see the Crofts every day when she was out. Captain Wentworth has not been hiding in a room since his arrival. He's out walking on two of the busiest streets: Milsom Street and Pultney Street.

Even if Lady Russell didn't see him either of those times, he has a party of friends in Bath who would certainly be talking amongst themselves about the fact that he had come.

We also know that Lady Russell "drinks the water, gets all the new publications, and has a very large acquaintance". She is right in the thick of everything that is going on, and those 'publications' also published the names of people who had arrived in the city. (I have a book that shows an actual scan of the Bath paper including Jane Austen's mother among the arrivals, and she had come only the day before the paper was published). There is also the fact that Mrs. Croft is Lady Russell's friend and would certainly have said something had they run into each other anywhere.

It seems to me that a person like Lady Russell would have to go out of her way to remain ignorant of Captain Wentworth's presence in Bath.